Dynamical modeling of infectious diseases serves as an important quantitative tool for understanding pathogen transmission patterns, evaluating the effectiveness of prevention and control measures, and predicting epidemic development trends. The compartmental modeling approach, owing to its clear structure and strong analytical tractability, has become the cornerstone of mathematical modeling in epidemiology. However, real-world infectious disease transmission often exhibits complex heterogeneity and multi-stage characteristics: different diseases have distinct incubation period features; public health interventions (e.g., isolation, quarantine) dynamically alter population contact patterns; and the clinical outcomes of infected individuals include both recovery and death. The classic SIR model, which crudely lumps all removed individuals into a single category and does not account for latent states or intervention mechanisms, struggles to accurately capture these real-world features.

To address modeling needs under different scenarios, researchers often introduce new compartments or adjust parameter structures based on the SIR model, constructing extended models of increasing complexity. However, as the number of compartments increases, the parameter dimensionality and data requirements also rise, leading to changes in model applicability, robustness, and interpretability.

This study focuses on five typical extended models derived from the classic SIR model [3]: the SI model, which includes only susceptible and infectious compartments; the SIR model, comprising susceptible, infectious, and recovered compartments; the SIQR model, which introduces a quarantined compartment to SIR; the SIQDR model, further distinguishing between recovered and deceased individuals; and the SEIQRD model, which simultaneously includes exposed (latent), quarantined, recovered, and deceased compartments. These five models exhibit a clear hierarchical and progressive relationship in terms of the number of compartments, assumptions, parameter scale, and applicable scenarios.

This paper aims to systematically outline the mathematical structures of the above five models, clarify their differential equation formulations and parameter definitions, and conduct a comparative analysis of each model in terms of modeling assumptions, applicable disease types, data requirements, and limitations. Through this study, we hope to provide a theoretical basis for infectious disease modelers in selecting and constructing models, as well as to lay the groundwork for subsequent parameter estimation and intervention evaluation based on real-world epidemic data.

Starting from the most basic SI model (which includes only infected and uninfected individuals), this paper successively introduces the recovered, quarantined, deceased, and exposed compartments. This approach allows the model to more closely reflect real-life conditions while maintaining accuracy and strong robustness against disturbances.

Table 1. Summary of model parameter symbols, units, and descriptions.

2.4.1. SI Model

1) Compartment Division:

S: Susceptible individuals

I: Infectious individuals

Total population N = S + I

2) Differential Equations:

2.4.2. SIR Model

1) Compartment Division

S: Susceptible individuals

I: Infectious individuals

R: Removed individuals

2) Differential Equations

2.4.3. SIQR Model

1) Compartment Division

S: Susceptible individuals

I: Infectious individuals

Q: Quarantined/Isolated individuals

R: Removed individuals

Total population N = S +I + Q + R

2) Differential Equations

2.4.4. SIQRD Model

1) Compartment Division

S: Susceptible individuals

I: Infectious individuals

Q: Quarantined/Isolated individuals

D: Deceased individuals

R: Recovered individuals

Total living population N living = S + I + Q + R + D , since deceased individuals are counted in the total living population, N living is no longer constant.

2) Differential Equations

2.4.5. SEIQRD Model

1) Compartment Division

S: Susceptible individuals

E: Exposed individuals

I: Infectious individuals

Q: Quarantined/Isolated individuals

R: Recovered individuals

D: Deceased individuals

Total living population N living = S + E + I + Q + R , since deceased individuals D are not counted in the total living population, N living is no longer constant.

R₀ is directly specified from reports; β is back‑calculated from the model’s R₀ formula in Table 1. β values in the subsequent simulations are effective per‑contact transmission probabilities.

2) Differential Equations

For model calibration and validation, we used publicly reported daily time series of new confirmed cases and new deaths for two cities:

Wuhan (2022): Data from Wuhan Municipal Health Commission official website [4], covering the period from January 1, 2022 to March 31, 2022. The dataset includes daily newly confirmed COVID-19 cases and deaths. The data required for the simulation are in Table 2 and Table 3, and the simulation results are shown in Figure 1.

New York City (2023): Data from the NYC Department of Health and Mental Hygiene (via NYC Open Data), covering the period from January 1, 2023 to June 30, 2023. Daily case counts and death counts were extracted [5]. The data required for the simulation are in Table 4 and Table 5, and the simulation results are shown in Figure 2.

Table 2. Initial parameters and basic reproduction number for COVID-19 in Wuhan (2022).

Initial conditions (simulation start date: January 1, 2022) ([7])

Table 3. Initial population compartments for Wuhan (2022).

Figure 1. SEIQRD model simulation results for the COVID-19 epidemic in Wuhan (2022).

The model simulation is based on the permanent resident population data of Wuhan in 2022 [7].

Table 4. Initial parameters and basic reproduction number for COVID-19 in New York City (2023).

Initial conditions (simulation start date: January 1, 2023) [7].

Table 5. Initial population compartments for New York City (2023).

Figure 2. SEIQRD model simulation results for the COVID-19 epidemic in New York City (2023).

The model simulation is based on the permanent resident population data of New York City in 2023 [8].

Given that Wuhan and New York City adopted markedly different epidemic control approaches (extremely strict in Wuhan, relatively lenient in New York City), there are also significant differences between the two major cities in terms of the number of infections and the cumulative death toll, which can be shown in Figure 3 and Figure 4.

To evaluate the extent to which key parameters influence the model outputs and to provide a quantitative basis for public health decision-making, this study conducts univariate and bivariate sensitivity analyses based on the baseline parameters of the 2023 COVID-19 epidemic in New York City. The selected parameters are the isolation rate δ , the isolation effectiveness coefficient k, and the basic reproduction number R 0 . The output indicators analyzed are the peak number of infected individuals I max and the cumulative number of deaths by the end of the epidemic D final . The results of these sensitivity analyses are presented in Figure 5.

To support the claim that the SEIQRD model outperforms the classic SIR model,

Figure 3. Comparison of the number of infected individuals.

Figure 4. Comparison of the number of deaths.

Figure 5. Univariate and bivariate sensitivity analysis of isolation rate, isolation effectiveness, and basic reproduction number.

we applied both models to the same COVID-19 datasets for Wuhan (2022) and New York City (2023), the results are shown in Table 6. The SIR model was implemented using the same total population and initial conditions. The SEIQRD model used the parameters reported in Sections 3.2 and 3.3.

Table 6. Comparison of fitting errors (RMSE) between SEIQRD and SIR models for Wuhan and New York City.

In both cities, SEIQRD reduces RMSE by more than 60% for infections and 70% for deaths, demonstrating substantially better alignment with observed epidemic dynamics.